PHYSICAL REVIEW FLUIDS 2, 064701 (2017)

Effect of radius of gyration on a wing rotating at low Reynolds number:A computational study

Daniel Tudball Smith,1,* Donald Rockwell,2 John Sheridan,1 and Mark Thompson1

1Fluids Laboratory for Aeronautical and Industrial Research, FLAIR, Department of Mechanical andAerospace Engineering, Monash University, Clayton, Victoria 3800, Australia

2Department of Mechanical Engineering, Lehigh University, Bethlehem, Pennsylvania 18015, USA(Received 21 September 2016; published 5 June 2017)

This computational study analyzes the effect of variation of the radius of gyration (rg),expressed as the Rossby number Ro = rg/C, with C the chord, on the aerodynamics of arotating wing at a Reynolds number of 1400. The wing is represented as an aspect-ratio-unityrectangular flat plate aligned at 45◦. This plate is accelerated near impulsively to a constantrotational velocity and the flow is allowed to develop. Flow structures are analyzed andforce coefficients evaluated. Trends in velocity field degradation with increasing Ro areconsistent with previous experimental studies. At low Ro the flow structure generatedinitially is mostly retained with a strong laminar leading-edge vortex (LEV) and tip vortex(TV). As both Ro and travel distance increase, the flow structure degrades such that athigh Ro it begins to resemble that of a translating wing. Additionally, the present studyhas shown the following. (i) At low Ro the LEV and TV structure is laminar and steady;as Ro increases this structure breaks down, and the location at which it breaks down shiftscloser to the wing root. (ii) For moderate Ro of 1.4 and higher, the LEV is no longer steadybut enters a shedding regime fed by the leading-edge shear layer. (iii) At the lowest Ro of0.7 the lift force rises during start-up and then stabilizes, consistent with the flow structurebeing retained, while for higher Ro a force peak occurs after the initial acceleration iscomplete, followed by a reduction in lift which appears to correspond to shedding of excessleading-edge vorticity generated during start-up. (iv) All rotating wings produced greaterlift than a translating wing, this increase varied from ∼65% at the lowest Ro = 0.7 downto ∼5% for the highest Ro examined of 9.1.

DOI: 10.1103/PhysRevFluids.2.064701

I. INTRODUCTION

In recent years, a rapid increase in research has resulted in increased understanding of low-Reynolds-number flight. Both the development of unmanned microaerial vehicles and advancementof computational and experimental techniques have driven this increased understanding. Theincredible performance of natural flyers at these smaller scales has pushed the study of flappingflight to the forefront of many of these investigations. Extensive reviews of flapping and biologicallyinspired flight have been given by researchers such as Ellington [1], Sane [2], Dickinson [3], Wang[4], and Shyy et al. [5]. In the context of the present study, Wolfinger and Rockwell [6] provide arecent review of investigations pertaining to rotational effects on wing performance.

The study presented herein investigates how the flow structures and performance of a rotatingwing develop and change with time by varying the contribution from rotational effects comparedto translational effects. This variation is achieved by analyzing an aspect-ratio-unity wing of fixedplanform and varying the radius of gyration (rg); that is, moving the wing closer to or furtheraway from the rotation axis. This has practical consequences for flapping unmanned aerial vehicledesign, where it is important to understand the effect of increasing rotational effects by shifting therotation axis. A better understanding of these circumstances is potentially important in effecting better

*daniel.tudball-smith@monash.edu

2469-990X/2017/2(6)/064701(23) 064701-1 ©2017 American Physical Society

TUDBALL SMITH, ROCKWELL, SHERIDAN, AND THOMPSON

designs of biomimetic flapping mechanisms as it would affect the design of the drive mechanism for alow-Reynolds-number flyer. For a wing rotating in a flow with no freestream velocity, representativeof a hovering flyer, the ratio of rotational effects to translational effects can be described by theRossby number, which for this case can be expressed as Ro = r/C, where r is the characteristicradius and C is the chord length [7]. This study therefore investigates the effect of Rossby numberthrough changes in radius of gyration, noting rg can be used as the characteristic length such thatRo = rg/C.

An understanding of rotating and flapping wings has been pursued in recent decades andsubstantial advances have been made. A number of researchers have sought to relate aerodynamicperformance to the wing kinematics of various flapping flyers [8,9]. Ellington et al. [10] employedqualitative flow visualization to characterize the flow structure along the wings of both live insectsand mechanical models, identifying the characteristic stably attached leading-edge vortex (LEV)of a rotating wing. Other researchers have employed quantitative flow visualization to characterizethe flow structure along the wings of live insects [11] and also dynamically scaled models [12–14].Forces on flapping wings have been measured by Lentink and Dickinson [7], Dickinson et al. [15],Birch and Dickinson [12], and Poelma et al. [13]. These have been supplemented by computationalstudies on flapping wings [16–20].

Studies such as Ellington [8] and Dickinson et al. [15] have suggested that flapping flight canbe generalized through several dynamic mechanisms: delayed stall as the wing sweeps through itsstroke, additional circulation generated by wing pitch variation during pronation and supination, andwake capture.

Though the full kinematics of natural flyers is complex and varies from one species to the next withgreat variation in the kinematics and wing geometries, a rotating wing that is accelerated impulsivelyto a constant rotation velocity is a common simplification used to analyze high lift mechanisms in themiddle portion of a wing stroke. Existing research has shown that this simplification of the kinematicsgives a good approximation of a downstroke in typical flapping flight and that high lift mechanismsassociated with flapping flyers also exist for this simplified case (e.g., [7,13], and various others). Theexistence of a stable and attached LEV on a rotating wing has been identified in many studies for arange of Reynolds numbers and wing geometries [7,13,21–29]). This stable low-pressure LEV is keyto the high performance of rotating wings, as it reattaches the flow separating from the leading edgeat high angles of attack that would otherwise cause a translating wing to stall [7,10,25,28]. Recentstudies have investigated the effects of low-aspect-ratio wings undergoing rotating and pitching[30–32] and rotating wings undergoing acceleration (surging) [30].

Various researchers have investigated the mechanisms responsible for a stable and attached LEV.Lentink and Dickinson [33] proposed, using an analytical approach, that centripetal and Coriolisforces, when observing the wing in a fixed reference frame, stabilize flow structures compared to atranslating wing. Numerical simulations [28,29] have demonstrated Coriolis forces on a fluid elementare small compared to forces due to pressure gradients, while centrifugal and pressure gradient forcesare responsible for spanwise flow and are important in retaining LEV attachment. Jardin and David’s[34] numerical study supported that of Lentink and Dickinson [33], where centrifugal and Coriolisterms were introduced into simulations independently and it was found that Coriolis effects have a keyrole to play in lift generation while the impact of centrifugal terms is only marginal. Jardin and David[35] compared the flow over a translating wing in uniform flow, a translating wing with a spanwisevelocity gradient as would be seen on a rotating wing, and a purely rotating wing. They found that thespanwise velocity gradient alone does not promote significant lift improvements like a rotating wingdoes, further emphasizing the importance of forces arising from a rotating reference frame. Addition-ally, Wojcik and Buchholz [36] proposed that vorticity annihilation due to a layer of opposite-signedvorticity on the plate surface was often a dominant mechanism for regulating the LEV. Recently,Limacher et al. [37] identified that Coriolis accelerations limit the spanwise extent of a stable LEVby tilting the LEV core towards the wake. If the average trajectory of the LEV is outwards (towardsthe tip) and away from the leading edge (towards the wake), then it follows that the correspondingaverage Coriolis force components will be towards the wake and towards the root, respectively.

064701-2

EFFECT OF RADIUS OF GYRATION ON A WING . . .

Experimental studies [26,38–42] and numerical studies [22,27,29] have further investigatedaspect-ratio effects of rotating wings. Harbig et al. [27] performed a numerical study on a scaledfruit-fly wing of varying aspect ratio and proposed scaling by a Reynolds number based on thewing’s span rather than its chord to decouple Reynolds number and aspect-ratio effects. This isfurther supported by Carr et al. [39]. Fu et al. [40] varied aspect ratio and found Kelvin-Helmholtzinstabilities in the LEV and vortex bursting at outer spanwise locations for higher aspect ratios.Kruyt et al. [41] showed that the leading-edge vortex of a rotating wing of high aspect ratio at highangle of attack remains attached closer to the root and separates beyond four chord lengths of radialdistance. Phillips et al. [42] estimated lift coefficients from LEV circulation, while Harbig et al.[27] estimated lift from numerical simulations. Both found that an increase in aspect ratio initiallyincreases the coefficient of lift and then decreases it at higher aspect ratios (AR � 5). Each ofthese aspect-ratio studies show degrading flow structure further from the rotation axis (more distantspanwise locations) for higher aspect ratios.

Wing performance has been linked with rotational effects, with qualitative and quantitativeflow visualization showing the difference between a wing rotating with Ro = O(1) (dominatedby rotational effects) and rectilinearly translating (Ro → ∞) [7,24,25,28]. As Rossby number isincreased the wing’s performance reduces [7,43]; this reduction in performance continues until thewing is in pure translation.

Wolfinger and Rockwell [6] provide a study of Rossby number effects for a fixed aspect-ratiowing where Ro is varied by changing rg . With flow field reconstructions using stereo particle-imagevelocimetry (SPIV), it shows that structures attributed to good performance, such as a strongleading-edge vortex pinned to the wing and a large downwash component, are present on wings atlow Rossby numbers but degrade as the Rossby number increases. Wolfinger and Rockwell [6] alsoshows on a rectangular AR = 1 wing for small travel distances at high Rossby numbers that the flowstructure can be initially favorable but deteriorates the further the wing travels. Therefore for lowRossby numbers, the favorable flow structures are preserved, with only minor degradation at verylarge travel distances (such that the wing has passed through several rotations).

Wolfinger and Rockwell [44] expanded on these findings by analyzing a larger aspect ratioAR = 2 wing of varying Ro during the start-up process. They found that for low Ro a strong LEVdevelops during start-up and persists. At high Ro, the LEV forms during start-up but is then shedinto the tip vortex after some time. Studies involving variation of aspect ratio [27,29,40,41] havefurther highlighted the need to understand the degradation of flow structures with increasing radialdistance, as well as the surface pressure distributions, and lift and drag at extreme values of Rossbynumber.

Unresolved Issues

For the case where the radius of gyration of a rotating wing having fixed (low) aspect ratiois systematically varied over a wide range, a number of aspects of the flow physics in relationto the forces acting on the wing have not been fully addressed. It is hypothesized that a numberof mechanisms contributing to the vortex physics may be important, but they have not been fullyidentified or explored in relation to the wide range of values of Rossby number and travel distance ofthe wing. These physical mechanisms are: transformation of the leading-edge and tip vortices from alaminar to a degraded state, and the location of such breakdown along the span of the wing; spanwisedistortion of the leading-edge vortex towards, for example, an archlike vortex; onset of shedding ofthe leading-edge vortex, as opposed to a stable leading-edge vortex; and the form, trajectory, anddegree of coherence of the tip and root vortices downstream of the wing, and the manner in whichthey either connect or remain unconnected within the near field of the wing.

These features of the flow structure will be intimately related to patterns of instantaneous surfacepressure coefficient that determine the overall loading of the wing. The possible occurrence ofregions of large suction pressure, as well as regions of positive pressure, as a function of Rossbynumber and travel distance, and their relation to the flow physics, remains unclarified. Moreover,

064701-3

TUDBALL SMITH, ROCKWELL, SHERIDAN, AND THOMPSON

time (τ0 = 0)τ0 τ1

Ω(t

)

0

Ωc

(a)

xy

z

t

C

rgro

Vrg

45◦

Φ

(b)

FIG. 1. (a) Velocity profile used for wing motion. (b) Schematic of the “wing.”

the degree to which the value of Rossby number influences the magnitude of the lift coefficient and,furthermore, how it changes the form of the lift coefficient as a function of travel distance, has notbeen addressed.

II. METHODOLOGY

A. Numerical method

The numerical method used in the present study has been adapted from that of Harbig et al. [27]using the commercial code ANSYS CFX version 16.2. The flow over the rotating wing is modeled bythe Navier-Stokes equations cast in a noninertial rotating frame of reference along with the continuityconstraint

∂ρu∂t

+ ∇ · (ρuu) = −∇p + μ∇2u − 2ρ(� × u) − ρ[� × (� × r)] − ρ�̇ × r (1)

and

∇ · u = 0. (2)

Here ρ is the density, u is the velocity in the rotating frame, p is the pressure, μ is the molecularviscosity, � is the rotational velocity vector, and r is the position vector. Spatial discretizationwas carried out using the formally second-order-accurate blend-factor scheme with β = 1, and asecond-order backward Euler scheme was used to integrate forward in time.

B. Geometry and motion

The geometry of the wing is similar to that used by Wolfinger and Rockwell [6] and is shown inFig. 1(b). A simplified model of a wing was used with rectangular planform, an aspect ratio of unity,and a thickness to chord ratio of 0.08 and is positioned at a geometric angle of attack of 45◦ androtated about an axis aligned with the y direction. The aspect ratio of the wing is calculated fromthe chord length and the span, not including the root cutout,

AR = (rt − rr )/C, (3)

where rt and rr are the tip and root radii of the wing.The wing is initially at rest in a quiescent fluid before being rapidly accelerated to a constant

rotational velocity, �c. The imposed angular velocity profile is defined by a piecewise polynomial fitto allow smooth transitions between different phases of the motion (stationary, accelerating, constantvelocity, and deceleration). In particular, it produces a motion profile that is continuous up to and

064701-4

EFFECT OF RADIUS OF GYRATION ON A WING . . .

FIG. 2. Mesh used for simulations. Left: The surface mesh that was swept away from the body to createa volume mesh. Right: A slice through the volumetric mesh showing the hexahedral mesh swept into twonear-field boxes and a tetrahedral mesh in the far field.

including the 2nd derivative of rotational acceleration. The motion profile is given by

�(t) ={(−20

(tτ1

)7 + 70(

tτ1

)6 − 84(

tτ1

)5 + 35(

tτ1

)4), t < τ1

�c, t � τ1.(4)

Here �c is the constant final rotational velocity and τ1 is the time at which that velocity is first reached.The coefficients of the polynomial were determined by imposing the conditions that �(0) = 0,�(τ1) = �c and that higher order derivatives are continuous at t = 0 and t = τ1. This produces themotion profile shown in Fig. 1(a). A value of τ1 was chosen such that the wing achieves maximumvelocity in one chord length of travel, while the value of �c was determined such that Re = 1400based on the constant velocity at the radius of gyration (Vrg = �crg) and the chord (C).

Initially, the wing was positioned so that the root radius was at ro = 0.62C, producing a Rossbynumber based on the radius of gyration [33] of Ro = rg/C = 1.2. A parameter sweep was conductedwith ro varied through 0.12C, 0.37C, 0.62C, 0.87C, 1.12C, 1.37C, 1.62C, 2.62C, 4.62C, and8.62C, corresponding to Rossby numbers of rg/C = 0.7,0.9,1.2,1.4,1.6,1.9,2.1,3.1,5.1, and 9.1,respectively. This set expands on those used in the experimental study of Wolfinger and Rockwell [6].An increase in radius of gyration represents a reduction of rotational effects relative to translationaleffects.

C. Mesh and time step validation

The simulation method is based on that used by Harbig et al. [27], although the method ofspatial discretization varied. Where Harbig et al. [27] used a predominantly tetrahedral mesh witha prismatic inflation layer on the wing, the simpler geometry used in the current study allows for apredominantly hexahedral near-field mesh to be used. The wing surface is divided into rectangularelements, as shown in Fig. 2. This is swept in all directions to create a volume of hexahedral elementsin the near field. The far field was discretized with tetrahedral elements, with nodes matching at theinterface between hexahedral and tetrahedral elements. Approximately 12 million elements wereused in this study (varying slightly with Rossby number) with a biased hexahedral region to increaseresolution near the wing corners, where element spacing was 0.0035C. Mesh independence wasverified by comparing predicted forces to those from a mesh of approximately 20 million elements;at Ro = 1.2 this resulted in less than a 0.4% difference in mean forces over the last four chordlengths of travel. A time step corresponding to 0.015C of wing movement per step was found tobe sufficient for convergence; halving the time step produced a variation of less than 0.2% in meanforces.

The computational domain is cylindrical in the same form as Harbig et al. [27], with a diameterof 34 wingspans (with the rotation axis at the center) and a height of 44 chord lengths (with the wingpositioned centrally in the vertical direction).

064701-5

TUDBALL SMITH, ROCKWELL, SHERIDAN, AND THOMPSON

rgΦ/C0 2 4 6 8 10

CL

0.6

0.8

1

1.2

1.4

1.6

1.8

rg /C0 2 4 6 8

CL

mea

n

0.6

0.8

1

1.2

1.4

1.6

1.8

LEV Bursting

LEV shedding

Near-symmetric flow

Laminar LEV/TV

translating wing

Ro = 0.7Ro = 0.9

Ro = 1.2

Decreasing Ro

Ro = 1.4Ro = 1.6Ro = 1.9Ro = 2.1Ro = 3.1Ro = 5.1Ro = 9.0

translating

FIG. 3. Lift force coefficient comparisons, with lift force normalized by dynamic pressure at rg and planformarea. Left: Time history for varying rg/C, plotted over the first eight chord lengths or 330◦ of travel, whicheveroccurs first. Right: Mean lift force, with the mean taken between 3 < rg�/C < 6 for Ro � 1.2 and 150◦ <

� < 300◦ for Ro = 0.7 and 0.9.

III. RESULTS

A. Time history of forces

The lift forces, shown in Fig. 3, and the pressure on the upper wing surface in Fig. 4 coincide withfeatures of the velocity field found by Wolfinger and Rockwell [6], with low pressures correspondingto vortical structures, while lift coefficient reduction at high Ro and travel distance correspond tothe deterioration of the LEV.

After being near-impulsively accelerated (over one chord length of travel), the forces rise for thefirst one to two chord lengths of travel, depending on Ro as the flow structure develops. For higherRo the lift coefficient peaks some time after the acceleration has ended and then decreases, whilefor the lower Ro the force first stabilizes and then continues to rise throughout the wing’s motionuntil the wing begins to encounter its own wake. The force peak is initially delayed as Ro increasesuntil Ro � 3.1, where it advances slightly. The initial delay of the force peak appears to be causedby differences in the shedding of the leading-edge start-up vortex. At low Ro this vorticity is shedinto the tip vortex soon after the acceleration has ended. However, as Ro increases up to 2.1, theleading-edge starting vortex initially lifts to form an arch and there is a larger separation regionassociated with the LEV. This allows this start-up structure a longer time to break down and becontained within this region before shedding and merging into the tip vortex. For Ro � 3.1 the forcepeak advances in time. This change in regime is due to the feet of the arch-type vortex lifting fromthe surface of the wing, reducing suction on the upper surface, while a second force peak occursbetween rg�/C of 4 and 5.5 due to this vortex bursting, followed by it being shed and merging intothe tip vortex. At the lowest Rossby numbers, Ro � 1.2, this starting vortex is better maintained sothat it does not lift to form an arch structure or breakdown in the same fashion; consequently,there is no force peak and reduction associated with the lifting and shedding of this start-upvortex.

The observation of an arch vortex is consistent with the AR = 2 results presented in Wolfingerand Rockwell [44], who found experimentally at higher Ro, the starting LEV forms an arch vortexthat develops relatively slowly and is then swept into the wake of the wing. The structure of the archvortex was first defined in the computations of Visbal [45] on a heaving wing; its structure is genericfor different wing motions and it has been confirmed experimentally on a heaving wing [46] and ona pitching wing with an inflow [47].

064701-6

EFFECT OF RADIUS OF GYRATION ON A WING . . .

Ro = 0.7 Ro = 1.2 Ro = 2.1 Ro = 5.1 translating

rgΦC = 0.7

rgΦC = 1.1

rgΦC = 1.9

rgΦC = 2.9

rgΦC = 4.2

rgΦC = 5.6

−3. 05 .5

TV

LEV arch

TV

LEV feet

LEV

LEV

TIP

RO

OT

FIG. 4. Instantaneous pressure coefficient on the upper surface of the wing for varying Rossby number andtravel distance.

It is noted by Lentink and Dickinson [7] that flying animals typically have a Rossby numberbased off the tip radius close to 3 (corresponding to Ro ≈ 2.5 as based on the radius of gyration inthe present study). Although aspect ratio also needs to be considered, this would correspond to aregion where the leading-edge start-up structures are better retained, creating a higher peak force.

At larger travel distances (rg�/C � 6 and/or � � 270◦), the lift force decreases for all cases.For higher Rossby numbers this decrease appears to be due to flow structure degradation, as thesystem is not capable of retaining stable high-leading-edge vorticity and favorable flow structuredevelopment during start-up, as was demonstrated by Wolfinger and Rockwell [6] for Ro = 5.1 andfor rg�/C � 10. For Ro � 1.2, however, lift decreases because the wing starts to encounter thedownwash it generated during start-up beyond � of 270.

064701-7

TUDBALL SMITH, ROCKWELL, SHERIDAN, AND THOMPSON

(a) (b)

FIG. 5. (a) Variation of mean lift force with both Rossby number and Reynolds number. Mean taken between3 < rg�/C < 6 for Ro � 1.2 and 150◦ < � < 300◦ for Ro = 0.7 and 0.9. (b) The Reynolds number functionas given in Eq. (7), plotted across the range of Re tested.

Figure 3 (right) shows the variation of mean lift force with Rossby number. In this case, the meanis taken over 3 < rg�/C < 6, except in the low-Rossby-number cases of Ro = 0.7 and 0.9, whichbegin to encounter their shed wake before rg�/C = 6. For Ro � 0.9 the average is instead takenover 150◦ < � < 300◦. Figure 3 (right) highlights the flow structure variation that occurs over therange of Ro tested.

Figure 5 presents the variation of mean force expanded to include a limited dataset at Re =350,700, and 2800 in addition to the Re = 1400 results already presented. It has been demonstratedby other researchers that in this Reynolds number range there is a general increase in lift forcefor increasing Reynolds number. Figure 5 shows that the combined effect of Rossby number andReynolds number can be simplified as a product of a function of the Reynolds number and a functionof the Rossby number:

CLmean = f (Re)f (Ro). (5)

The Rossby number function is determined by the specific geometry and how the Rossby number isdefined (in this case by the radius of gyration). Empirically, for the given dataset it has been found that

f (Ro) = A

Ro0.5+ B, (6)

that is, that the lift force is a function of 1/Ro0.5. The constants A and B (which are dependenton the specific geometry) in this case are determined to be roughly equal to each other, so that theequation for CLmean can be expressed as

CLmean = f (Re)

(1

Ro0.5+ 1

). (7)

The implication is that Rossby and Reynolds number effects on the mean force trends can bedetermined separately and combined.

It should be noted that the empirical function f (Ro) applies only to the intermediate range ofRossby numbers, as low Ro results deviate from this trend and the asymptote for high Ro impliedby the equation is lower than CLmean for the translating wing case. For low Rossby number thisis not unexpected, as the mean force is taken over a reduced period where the force appears to becontinuing to rise before the wing encounters its own shed wake at � � 270. For exceptionally high

064701-8

EFFECT OF RADIUS OF GYRATION ON A WING . . .

Ro = 0.7 Ro = 1.2 Ro = 2.1 Ro = 5.1

rgΦC = 0.7

rgΦC = 1.1

rgΦC = 1.9

rgΦC = 2.9

rgΦC = 4.2

rgΦC = 5.6

root

LEV

A

B C

D

E

FIG. 6. Development of flow structure as Rossby number and travel distance rg�/C increase. Isosurfacesplotted at QC2/V 2

rg = 3.

Ro numbers greater than those tested in the present study, flow likely stabilizes to the translatingcase rather than continuing to follow the outlined trend.

B. Flow structure development: Pressure and vortex structures

Figure 6 shows the development of flow structures visualized by an isosurface of Q criterion forvarying Rossby number, and also the time development within a run for each Rossby number. It

064701-9

TUDBALL SMITH, ROCKWELL, SHERIDAN, AND THOMPSON

is useful to consider this in conjunction with Fig. 4, which shows the development of the pressuredistribution on the upper surface of the wing.

The initial starting vortex structure (travel distance of the wing, rg�/C = 0.7) is similar for allcases, with vortices forming on all edges of the wing. The tip and trailing-edge vortices have alreadybegun lifting away from the surface (A in Fig. 6) while the root and leading-edge vortices are stillattached, with the leading-edge vortex being conical in structure. Initially the pressure distributionsare similar for each of the rotating-wing cases (Ro from 1.2 to 5.1), with the LEV for lower Ro casesgiving a wider region of low pressure near the leading edge.

By rg�/C = 1.1 differences become apparent—the outer portion of the leading-edge vorticitythat forms during start-up cannot remain stably attached to the wing, and it starts to lift away fromthe upper surface for higher Rossby numbers (Ro � 1.2). For Ro of 1.2 this occurs only near the tipof the wing (B in Fig. 6); however, the higher Rossby numbers do not sustain the LEV as well and alarge portion of the LEV lifts away from the surface as an arch (C in Fig. 6). This is also seen in thesurface pressure, where for higher Rossby numbers a gap is present in the region of suction causedby the LEV lifting away with low-pressure regions at the LEV feet (Fig. 4). At rg�/C = 1.9, thelower Rossby number cases have started to shed excess leading-edge vorticity from start-up intothe tip vortex (TV), highlighted by D in Fig. 6. The shedding of this excess leading-edge vorticitycorresponds to a stabilization of the lift force for the lowest Ro cases (�1.2) and the force peak athigher Ro (1.4 � Ro � 2.1) observed in Fig. 3.

By this time all Rossby numbers have shed two vortices from the trailing edge highlighted by Ein addition to the initial starting vortex. As the flow structures develop in time, the tip vortex andwake structures are increasingly disordered for higher Rossby numbers, in spite of having the sameReynolds number based on chord length.

Even at small travel distances, the translating wing (Ro → ∞) shows significant differences fromthe rotating wings, with larger regions of lower magnitude suction. These correspond to the feet ofan arch-type leading-edge vortex as described by Garmann et al. [28], which lifts at the center. Theregions of suction nearest the tips of the translating wing are the feet of the tip vortices that arelifting from the surface. For the translating wing, the lift force reaches a peak at rg�/C = 1.7, justbefore the arch-type leading-edge vortex lifts entirely away from the upper surface of the wing. Asmentioned previously, this arch vortex is also noted on the AR = 2 wing of Wolfinger and Rockwell[44]. The high Ro structures seen in Fig. 6 and the associated force trace resemble the AR = 1translating wing case of Taira and Colonius [48], in spite of the lower Re (300) and angle of attack(30◦) of that case.

Figure 7 shows (a) the spanwise velocity and (b) the spanwise vorticity flux (defined as wrωz,where wr is the spanwise velocity in the relative frame and ωz is the spanwise vorticity component).These figures are plotted at an instant in time after the forces have stabilized (i.e., after theinitial shedding of start-up structures and the arch vortex). Figure 7(a) demonstrates that as theRossby number increases there is a drop in the spanwise velocity component through the LEV(visualized with isolines of Q criterion). At Ro = 1.4 and 0.50 span the spanwise velocity throughthe LEV drops rapidly as vortex breakdown is encountered. At Ro = 1.6 and 0.50 span a lowerintensity vortex with a positive (though much reduced) spanwise velocity is reestablished, asthe core of the LEV begins to reform further from the leading edge and higher off the wing’ssurface.

Figure 7(b), showing wrωz, gives some indication of the spanwise flow of spanwise-orientedvorticity. This will have some magnitude where spanwise vorticity and spanwise velocity arecoincident. It is clear that through the core of the vortex, the spanwise vorticity flux is greaterfor lower Rossby numbers. The vorticity flux drops to near zero where the LEV bursts for higherRossby numbers, though recovers a slight positive value closer to the tip as the LEV is reestablishedbefore being drawn in to the tip vortex, though at a much reduced intensity compared to the lowerRossby number cases. Additionally shown is the buildup of negative vorticity flux in the shear-layerregion of the LEV. This negative value is due to slight negative velocity as the shear layer is slightly

064701-10

EFFECT OF RADIUS OF GYRATION ON A WING . . .

Ro = 0.7

Ro = 1.2

Ro = 1.4

Ro = 1.6

0.25 span 0.50 span

(a)

(b)

0.75 span

wrωzCV 2

rg

75

−75

Ro = 0.7

Ro = 1.2

Ro = 1.4

Ro = 1.6

0.25 span 0.50 span 0.75 span

wVrg

1.6

−1.6

FIG. 7. Spanwise planes taken at 0.25, 0.5, and 0.75 span for varying Ro. (a) Spanwise velocity above theupper surface of the wing. (b) Spanwise vorticity flux wrωz. Plots are overlaid with isolines of Q criterion tovisualize the vortex structures.

deflected toward the root of the wing in combination with a very large positive vorticity from theshear layer itself.

Pressure fields on the same spanwise slices are presented in Fig. 8. As would be expected, thepressure is lower in the region of the LEV core where vorticity is highest and the suction at the LEVcore reduces as the Rossby number increases. This leads to the pressure distributions already shownpreviously in Fig. 4. Additionally of note is that the entire separated region and not just the core ofthe LEV has a much reduced pressure for lower Rossby number.

064701-11

TUDBALL SMITH, ROCKWELL, SHERIDAN, AND THOMPSON

Ro = 0.7

Ro = 1.2

Ro = 1.4

Ro = 1.6

0.25 span 0.50 span 0.75 span

Cp

0

−8

FIG. 8. Spanwise planes taken at 0.25, 0.5, and 0.75 span for varying Ro showing the pressure distributionnear the wing.

C. Detailed LEV structure: Particle tracking and planes of Q criterion

To better understand the behavior of the LEV as Ro is varied, particles were released along theleading edge of the wing; these are shown in (Fig. 9) along with planes of Q criterion (Fig. 10). Atthe lowest Ro = 0.7, there is a strong and coherent LEV originating from the root of the wing allthe way to the tip, with only a slight deviation at the transition between the LEV and the TV. Theparticle tracking shows laminar flow throughout the LEV and TV evolution. At Ro = 1.2 the sameconical LEV is observed; however, there is a sudden expansion around the midspan of the wingcharacteristic of vortex bursting. The particle trace shows that the flow becomes disordered in the

FIG. 9. Particle tracking for a range of Rossby numbers at a travel distance rg�/C = 5.5. See SupplementalMaterial [51] for a video representation of particle tracking.

064701-12

EFFECT OF RADIUS OF GYRATION ON A WING . . .

FIG. 10. (top) Slices of Q criterion (bottom) midspan isolines of Q criterion. All results in this figure aregiven at a rotation angle � = 230◦. NOTE: To make visualizations clearer, the spanwise direction has beenscaled by a factor of 2.

transition between the LEV and the TV. These effects result in a decrease of the LEV core strengthnear the tip of the wing, as seen in Fig. 10.

For Ro = 1.6 and higher the LEV undergoes further changes, with an increasingly larger volumeof particles becoming entrained in the larger separated region between the LEV and the upper surfaceof the wing as the LEV becomes less coherent. In this range of Ro the LEV ceases to remain astable vortex and transitions to periodic shedding. The region highlighted for Ro = 2.1 in Fig. 10(A)shows a buildup of rotational flow fed by the leading-edge shear layer, which sheds into the regionof higher Q immediately downstream. The onset time of this LEV shedding varies slightly withRo, for higher Ro occurring at rg�/C � 5.5 after the flow has settled from shedding the excessleading-edge vorticity from start-up. This shedding occurs closer to the tip (>0.3 span for Ro = 2.1),while closer to the root the LEV does not shed (<0.3 span), and in the region between shedding andnonshedding regions (∼0.3 span) the LEV is dominated by shear, making identifying a vortex coredifficult.

More general trends as Ro increases are that the LEV is seen to be less intense and forms higherand further back from the leading edge and wing surface, as seen in Fig. 10 for Ro = 0.7, 1.2,and 2.1. This leads to the lower suction observed in Fig. 4 and an increasingly larger recirculationregion that is seen to entrain a larger volume of particles in Fig. 9. This results in the overallreduction in performance, seen through the force history and mean forces presented in Fig. 3.This lifting and weakening of the LEV has been observed nearer the tip of high-aspect-ratio wingsin previous aspect-ratio studies [27,29,39–42]. However, this study, along with the experimentalflow visualizations of Wolfinger and Rockwell [6], shows a similar trend of degradation, even forlow-aspect-ratio wings at higher Rossby numbers. These will directly affect the wing’s performance.

IV. VORTEX CORE PARAMETERS

The health of the LEV can somewhat be characterized by the flow parameters through the coreof the vortex. For this purpose the center of rotation of the LEV is determined using the vortexidentification parameter γ1, as described by Graftieaux et al. [49]. This parameter (γ1) is a scalarfield that is a measure of the rotation around each point in the flow over a defined window. It isgiven by

γ1(P ) = 1

N

∑S

(PM ∧ UM ) · z

||PM|| ||UM || = 1

N

∑S

sin(θM ), (8)

064701-13

TUDBALL SMITH, ROCKWELL, SHERIDAN, AND THOMPSON

FIG. 11. (a) LEV center of rotation determined from γ1 (left) as viewed perpendicular to the wing uppersurface and (right) viewed front on, perpendicular to the leading edge and parallel to the upper surface.(b) Plots of axial velocity, axial pressure gradient, and vorticity magnitude through the center of rotation of theLEV determined from γ1.

where N is the number of points M inside the domain S that surrounds the point of interest P , andUM is the velocity at the point M .

A peak in γ1 therefore defines a center of rotation. In this case, the relative frame velocity hasbeen used to determine the center of rotation as the LEV is mostly fixed within the wing’s referenceframe. The position of the center of rotation as determined by γ1 is provided in Fig. 11(a). It can beseen that the position of the center of rotation of the LEV initially moves further down and awayfrom the leading edge of the wing as Ro increases. Between Ro = 1.2 and Ro = 1.4 the LEV breaksdown, and the center of rotation is disturbed between 0.3 and 0.4 span before forming again furtheraway from the leading edge at span >0.4. The projected frontal view shows that the center of rotationis also further from the surface of the wing.

Figure 11(b) then shows various parameters plotted at the center of rotation: the axial velocity,the axial pressure gradient, and the vorticity magnitude. The axial velocity shows that nearer theroot, the higher Ro has higher axial velocity, however, it is short lived, while lower Ro gains and thenmaintains axial velocity better as the LEV gets closer to the tip. All but the lowest Ro suffer a majordrop in axial velocity to near-zero magnitude associated with vortex breakdown. Axial velocity forthese cases picks up again further along the span as the LEV is reestablished.

The axial pressure gradient shows similar overall trends, with negative values indicating afavorable pressure gradient. It is seen that the higher Rossby numbers (Ro � 1.9) do not establishan appreciable favorable pressure gradient through the center of rotation of the LEV, only becomingslightly favorable nearer the tip as the pressure drops, leading into the tip vortex.

The vorticity magnitude through the center of rotation is seen to increase with span untilapproximately the midspan for the low Ro cases. By comparison, moderate Ro sees only a smallincrease near the root before dropping, while for high Ro, vorticity magnitude drops immediatelyfrom the root of the wing.

064701-14

EFFECT OF RADIUS OF GYRATION ON A WING . . .

ωzCVrg

-50

50

Ro = 2.1Ro = 1.6Ro = 1.4

Ro = 1.2Ro = 0.9Ro = 0.7

FIG. 12. Midspan spanwise vorticity plot for varying Rossby number. Isoline of Q = 0 shown with theblack line, region used for the circulation calculation shown by the blue box.

V. SPANWISE CIRCULATION OF LEV

Figure 12 shows midspan plots of spanwise vorticity for a range of Rossby numbers; also plottedare an isoline at Q = 0 and a bounding box highlighted in blue. The bounding box shown in thisfigure was used as the region for calculation of the spanwise circulation of the LEV; the choice ofbox size is a compromise such that it contains the more dispersed LEV structures for higher Rossbynumber (requiring a larger box), while avoiding capturing the tip vortex for low Rossby numbers(requiring a smaller box). Figure 12 demonstrates the trend of reduction in strength and lifting ofthe LEV core for increasing Ro.

Figure 13 shows the spanwise circulation, �z/(VrgC), on the upper side of the various wings. Thespanwise circulation calculated over the bounding box shown in Fig. 12 is presented in Fig. 13 (topleft), described as the “loose” bounding box, while Fig. 13 (middle right) shows the circulation over arectangular bounding box more tightly fitted to the LEV, sized to capture the region of the LEV whereQ > 0, as shown in Fig. 12. The isoline of Q > 0 was not used in these circulation calculations,as multiple disconnected loops are present in the LEV region when the LEV undergoes breakdownat higher Ro, especially nearer the tip. Both of these plots demonstrate a trend of increasingspanwise circulation in the LEV with span, while the loose box shows lower Ro wings have lowerspanwise circulation on the upper surface of the wing nearer the root but higher circulation nearerthe tip.

There are various relevant factors that contribute to the behavior. First, from a vorticity generationperspective, the lower Rossby numbers have a lower root speed and higher tip speed, thus generatingless circulation from leading-edge separation near the root and more nearer the tip. Second, froma vortex geometry perspective, for low Ro LEV circulation is swept outwards more rapidly andreaches closer to the tip before being reorientated into the wake. The drop off in spanwise circulationat larger spanwise locations is due to vorticity reorientating itself into the wake. These resultscan be compared with higher-aspect-ratio studies of Phillips et al. [42] that showed a generalincrease in circulation between 0.5 and 0.75 span, and Carr et al. [39], which showed reductionsin LEV circulation for aspect-ratio 4 compared to aspect-ratio 2 wings when normalizing bythe span.

Figure 13 (bottom) shows the spanwise circulation of the LEV integrated across the span of thewing for each Rossby number [i.e., the integral of the curves in Fig. 13(a) plotted against Ro]. Whenconsidering the looser box, which better represents the circulation over the entire upper surface ofthe wing, the total circulation is relatively constant across the range of Ro presented. Consideringthe tighter box, which better represents the circulation only in the LEV region itself, shows a slightincrease in total circulation with increasing Rossby number. These trends highlights that it is notthe circulation of the LEV alone which contributes to lift generation, but rather its location, peakintensity, and ability to rapidly reattach the flow that separates from the leading edge.

064701-15

TUDBALL SMITH, ROCKWELL, SHERIDAN, AND THOMPSON

tight boxloose box

Decreasing Ro

Decreasing Ro

Decreasing Ro

FIG. 13. Spanwise circulation of LEV, �z/(VrgC), (top left) at varying spanwise location from root to tipwhen calculated over the loose bounding box shown in Fig. 12, (top right) calculated over a tighter rectangularbox, sized to only include the LEV where Q > 0, and (bottom) the circulation averaged across the span for arange of Ro (rg/C).

VI. COMPONENTS OF DYNAMIC FORCE

As previously highlighted, the relative contribution of Coriolis and centrifugal forces to the totaldynamic force is still not well understood. Although the current study cannot unravel the complexitiesinvolved, some trends are highlighted in Figs. 14 and 15.

Figure 15 shows the rotational inertial forces at the point identified as the LEV core (based onthe peak in Q criterion on spanwise-oriented planes). This demonstrates the relative direction andmagnitude of the Coriolis and centrifugal forces at the vortex core, highlighting their decrease as

0.5 Span0.3 Span

FIG. 14. Pressure gradient, Coriolis, and centrifugal forces averaged across LEV, (left) at 0.3 span and(right) at 0.5 span.

064701-16

EFFECT OF RADIUS OF GYRATION ON A WING . . .

Ro = 0.7 Ro = 0.9 Ro = 1.4

Tip

Roo

tz

x

FIG. 15. Centrifugal and Coriolis forces at the leading-edge vortex core (peak Q criterion) for Ro = 0.7,0.9, and 1.4.

Rossby number increases. By contrast, the LEV forms closer to the leading edge as these forcesdecrease.

Figure 14 shows the spanwise components (z direction) of the pressure gradient, Coriolis andcentrifugal forces, and also the streamwise (x direction) component of Coriolis force is provided.These force contributions are averaged over the area of the leading-edge vortex bound by Q > 0(as shown in Fig. 12). The graphs are plotted such that positive values are considered favorable tomaintain a stable LEV, with forces towards the tip or towards the leading edge being positive, whilenegative forces are unfavorable, i.e., towards the root or towards the wake. The rotational referenceframe forces are expressed in their inertial form.

As the bulk of the spanwise and streamwise velocity in the LEV is towards the tip and towardsthe wake, it follows that the Coriolis force components must be towards the wake and towards theroot, respectively. This is, of course, consistent with the recent study by Limacher et al. [37] thatidentified that Coriolis accelerations limit the spanwise extent of the LEV. While Garmann et al.[28] looked at the detailed force component distributions, the aim here was rather to quantify theeffects only in an integrated sense.

The change in force component gradients for the 0.3-span case at Ro = 1.4 arise from the flowstructure changes described previously; the LEV goes from being a singular structure to a less stabletime-varying structure. It must be highlighted that Fig. 14 represents averages over the entire LEV;when considered only at the vortex core the spanwise (or axial) pressure gradient is adverse nearregions of vortex bursting for Ro � 0.9. Clearly the results indicate forcing along the wing towardsthe tip for the pressure gradient and centrifugal components, and towards the root for the Corioliscomponent, with the overall magnitudes of these competing components decreasing with Ro, asshould be expected. Jardin and David [34] highlighted the importance of Coriolis accelerations ingeneration of lift; however, it is not clear from the present study the mechanism(s) by which thisoccurs.

The pressure gradient force presented in Fig. 14 is somewhat abstract in that it gives an idea of therelative magnitude of the force in the LEV region, but it is difficult to interpret the actual effect onthe LEV. Similarly, the pressure gradient at the vortex core itself has not been presented in Fig. 15,as it will always be positive in radial directions transverse to the vortex axis, and in the directionalong the vortex axis, representative of core stability (as presented earlier in Fig. 11). To the endof better understanding the pressure gradient force, presented in Fig. 16 is the pressure plotted onlines which pass through the vortex core. Figure 16 has been broken down into: (left) a line plottedfrom the leading edge through the vortex core, as this is the direction of force pulling the LEV

064701-17

TUDBALL SMITH, ROCKWELL, SHERIDAN, AND THOMPSON

FIG. 16. Pressure plotted on a line starting from the wing surface and passing through the LEV core at thespecified spanwise locations, Z. (left) A line starting from the leading-edge corner. (right) A line starting fromthe upper surface of the wing and projected perpendicular to the wing’s surface. (top) Low-Rossby-numbercases and (bottom) high-Rossby-number cases after the LEV has broken down.

towards or away from the leading edge; and (right) a line plotted perpendicular to the wing’s uppersurface, originating at the wing’s surface and passing through the LEV core, as this is the directionpulling the LEV towards or away from the wing’s surface. Figure 16 has also been broken down byRossby numbers, with (top) low Rossby numbers before LEV breakdown and (bottom) high Rossbynumbers after vortex breakdown and reestablishment. The low Ro number cases have been plottedat different spanwise locations, such that the distance from the LEV to the leading edge is the same,while high Ro number cases are all plotted at the same spanwise location of 0.7 span, as there islittle difference in the location of the LEV core for high Ro cases.

The local minima seen in Fig. 16 are at the location of the vortex core, and it can be observedthat the suction at the vortex core decreases with increasing Rossby number, as can be observed ontwo-dimensional planes presented in previously in Fig. 8. However, of interest in Fig. 16 is that thepressure is lower not only at the vortex core itself, but also lower in the region between the vortexand either the wing surface or the leading edge. This is more noticeable in the high-Rossby-numbercases on the line plotted from the leading edge (i.e., the bottom left plot in Fig. 16), as there ismore space between the LEV core and the wing where the pressure recovery can occur, but is alsovisible in the low-Rossby-number and high cases. This difference in pressure between either sideof the LEV core results in a greater force on the LEV core itself, pulling the core both closer to thewing surface and also closer to the leading edge. This contrasts with the Coriolis force presented inFig. 15, as the Coriolis force approximately acts in the opposite direction to the force created by apressure gradient as presented in Fig. 16.

VII. DYNAMIC FORCE DISCUSSION

When examining the effect of Rossby number by itself, three additional major effects can beconsidered on a rotating wing compared to a purely translating wing affecting the force on a fluidelement near the wing: the Coriolis force, the centrifugal force, and a tangential velocity gradient inthe spanwise direction. The centrifugal and Coriolis effects are included in the momentum equation

064701-18

EFFECT OF RADIUS OF GYRATION ON A WING . . .

used for simulations [Eq. (1)] to account for being in the wing-stationary noninertial rotatingreference frame.

These inertial forces which exist in the wing’s reference frame approach zero as the Rossbynumber approaches infinity (i.e., a translating wing) and are given by

Fcent = −ρ[� × (� × r)] (9)

for the centrifugal force, and

FCor = −2ρ(� × u) (10)

for the Coriolis force. This demonstrates that the inertial centrifugal force is always acting outwardswhile Coriolis force is dependent on u, the local relative frame velocity. If the fluid element is fixedwithin the wing’s reference frame or only in the direction parallel to the rotation axis, the Coriolisforce will be zero. For an outward spanwise velocity, the inertial Coriolis force acts to draw thefluid element away from the leading edge and towards the wake (positive x direction), while fora tangential velocity away from the leading edge the Coriolis force is towards the rotation axis(opposing the centrifugal force).

Other than the leading-edge shear layer, which is typically deflected slightly rootwards in thepresent study, these are the primary flow directions seen at and near the LEV core, such that theCoriolis effect acts mostly to pull the LEV away from the leading edge and towards the root. Thisresult is highlighted in recent studies by Garmann et al. [28], Limacher et al. [37]. These rotationalframe inertial forces—centrifugal and Coriolis—vary with 1/Ro, as highlighted by Lentink andDickinson [7], so long as it is a correct assumption that the relative frame velocities that define theCoriolis force vary with the tangential velocity of the wing.

The other effect of wing rotation that can affect the forces is the tangential velocity gradient inthe spanwise direction, as this leads to a gradient in the generated leading-edge vorticity, as it rollsup from the separating shear layer. The difference in the circulation along the wing generates aspanwise pressure gradient. As rotational velocity �C in the presented simulations has been set suchthat the velocity at the radius of gyration Vrg is constant across simulations, it therefore implies thetangential velocity gradient also varies with 1/Ro.

Due to each of these rotating effects—the inertial forces and the spanwise tangential velocitygradient—varying with 1/Ro, it is not possible to separate the effects by observing a trend in Roalone.

Lentink and Dickinson [7] presented that it was not the inertial centrifugal and Coriolis forcesin and of themselves that stabilize the LEV but rather a reaction to them (i.e., an inward centripetalacceleration reacting against the outward centrifugal force and an outward or upstream accelerationin reaction to the inward or leeward inertial Coriolis force). For the accelerations presented in Lentinkand Dickinson [7] to exist, the inertial rotational forces require a reaction, which can be exertedthrough the wing itself. Lentink and Dickinson [7] presented that this may be through frictionalshear forces acting through the fluid onto the wing and was likened to Ekman pumping. Jardin andDavid [34] presented numerical simulations with and without the Coriolis force to show that theCoriolis force needs to be present to stabilize the LEV, preventing lifting and/or breakdown of theLEV and allowing the wing to produce more lift for a longer period of time.

The current study presents that a greater magnitude inertial Coriolis force pulling the LEV coreaway from the leading edge exists when the wing is positioned closer to the rotation axis. Thesecases also have the highest lift force and have the greatest attachment of the LEV core to the leadingedge. These lower-Rossby-number cases also have a significantly lower pressure, not only at thevortex core itself, but in the separated region immediately upstream and beneath the LEV core. Thisdemonstrates that as Rossby number decreases, the Coriolis force increases in a manner to pushspanwise moving fluid of the LEV away from the leading edge and into the wake; however, thedeveloped pressure distribution in the separated region is associated with pressure differences acrossthe LEV that act to pull it towards the leading edge and the wing surface. This still, however, doesnot form a complete picture, as it only considers bulk forces on fluid elements in the region of the

064701-19

TUDBALL SMITH, ROCKWELL, SHERIDAN, AND THOMPSON

Ro = 1.2 Ro = 2.1 Ro = 5.1

Exper

imen

tal

Num

eric

al

-10

10

ωzCVrg

FIG. 17. A comparison of numerical and experimental results. Spanwise planes of spanwise vorticity plottedat the midspan of the wing at a travel distance rg�/C = 5.5 for varying Ro. Experimental results are SPIVresults using a nine-image set average, reproduced from Wolfinger [50].

LEV after it has reached a semiequilibrium state, likely generated through nonlinear interactions. Inaddition, localized effects may contribute to LEV stability in a way that is not immediately obviousor separable from other effects.

VIII. EXPERIMENTAL COMPARISON

Figure 17 shows a comparison of numerical results of the current study alongside experimentalresults from Wolfinger [50]. For this comparison, the numerical results have been reduced inresolution using a technique similar to that outlined in Garmann and Visbal [29], with resultsaveraged across virtual interrogation windows with overlap to produce a vector spacing the same asthat of the SPIV results. Spanwise vorticity (ωz) was then calculated using central differencing inthe same manner as the published experimental results.

The main disparities between the experimental and numerical results are first that the numericalresults exhibit a slightly higher magnitude of vorticity in the LEV, the TV, and also the shear layersalong the wing’s surface. Second, the numerical results show the presence of positive vorticity justbehind the negative vorticity of the TV. This positive vorticity is the result of a vortex structureoriginating from the tip of the wing, fed by the positive vorticity of the shear layer on the top surfaceof the wing. This shear layer breaks down at higher Ro, resulting in the vortex being less ordered forsuch cases. There is a small concentration of vorticity in the same location in the experimental resultsfor Ro = 1.2; however, it is of much lower strength and it is not clear if this is the same structure.The third discrepancy is in some of the smaller structures not lining up or not being present in theexperimental results. This is somewhat to be expected, as the numerical results are an instantaneoussnapshot of the flow structure while the experimental results are reconstructed from nine SPIV imagesets phase-locked with the wing’s motion. The results of the present study have shown that flowstructures for higher Ro are unsteady such that in an averaged visualization they may appear as alarger diffused structure rather than small discrete structures if they are not perfectly in phase withthe wing’s rotation. Importantly, the key finding that the LEV weakens and moves further away fromthe wing’s surface is a consistent feature of both the experimental and numerical results, with thedominant flow structures located in the same positions relative to the wing.

IX. CONCLUSIONS

The results presented herein show that for a wing rotating at constant velocity the performanceof the wing degrades as the Rossby number increases. Here, the Rossby number increase is effected

064701-20

EFFECT OF RADIUS OF GYRATION ON A WING . . .

by increasing the radius of gyration. At small Rossby numbers (Ro � 2) the mean lift coefficientchanges rapidly with Ro, but by Ro � 9 the lift coefficient asymptotically approaches the value fora purely translating wing (Ro = ∞).

As the wing starts rotating, its wake structures at different Ro are similar. However, after theacceleration phase the wake structures evolve differently. In particular, the crucial LEV, whichremains stable and attached at low Ro, at higher 1.2 � Ro � 1.6 bursts, causing it to expand andfeed into the tip vortex. Consequently, there is a gradual decline in lift coefficient with time (rotationangle). For 1.6 < Ro � 2.1, the lift force increases during acceleration and then continues to rise toa peak before decreasing; this peak is slightly delayed with increasing Rossby number. This resultsfrom the LEV lifting off from the wing, forming an arch that subsequently sheds into the tip vortex.For higher Ro, the drop off after the peak in lift is more rapid with time. This appears to be due tothe LEV formed during start-up lifting and breaking down abruptly before being shed into the tipvortex.

The consequences of these different Ro wake evolutions with time on the lift coefficient arisefrom the formation and evolution of the vortex structures. Their influence on the surface pressuresof the wings is clearly seen and explains how the force histories evolve.

An understanding of how the LEV, the primary lift-producing vortex structure, is affected by thebalance between Coriolis and inertial forces is clearly important. This work adds to the existingunderstanding by directly linking the changes in wake structure to the lift forces via the inducedpressure fields. It also provides information on the dynamics of the lift generation, which could alsobe useful to designers of oscillatory rotating-wing devices.

ACKNOWLEDGMENT

A generous computing resource allocation through the National Computational Infrastructure(NCI) under Project No. d71 is gratefully acknowledged.

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[51] See Supplemental Material at http://link.aps.org/supplemental/10.1103/PhysRevFluids.2.064701 for videorepresentation of particle tracking over a range of Rossby numbers.

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